For Ns = 1, where Ns is the number of streams per user, the beam steering vectors are calculated by finding the array response vectors corresponding to the largest effective channel gain, i.e, finding the path that maximizes abs(abs(Ar(:,r)'*H*At(:,t))) where Ar and At are functions of the receive and transmit antenna array responses, respectively.
My question is, how to calculate the beam steering matrices for the case in which Ns> 1?
I think that you need to use information about the number of users and angle coordinates of all users in MIMO network.
Vadym Slyusar Thanks for your reply. Could you please explain this/how a bit more?
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In general, when Ns > 1, beamforming can be achieved by using a linear combination of beam steering vectors. The optimal linear combination can be found using a technique called singular value decomposition (SVD).
Specifically, if the channel matrix H is an Nt x Nr matrix, where Nt is the number of transmit antennas and Nr is the number of receive antennas, we can perform SVD on H as follows:
H = U * S * V'
where U is an Nt x Nr unitary matrix, S is a Nr x Nr diagonal matrix, and V is an Nr x Nr unitary matrix.
The diagonal entries of S represent the singular values of the channel matrix, which can be interpreted as the "strength" of the channel. The columns of U and V are the left and right singular vectors, respectively, which represent the directions of maximum channel strength.
To perform beamforming for Ns streams, we can select the first Ns columns of U to form a matrix U_Ns. We can then define the beam steering matrix as:
W = U_Ns'
This means that the transmit signal is multiplied by the conjugate transpose of the first Ns columns of U. The resulting signal is transmitted using the antenna array.
At the receiver, we can perform maximum ratio combining (MRC) to combine the received signals from different antennas. The MRC weight vector can be found by taking the conjugate transpose of the first Ns columns of V, as follows:
w_MRC = (V(:,1:Ns))'
This means that the received signals are multiplied by the conjugate transpose of the first Ns columns of V, and the resulting signals are added together to form the final output signal.
In practice, hybrid beamforming is often used in massive MIMO systems, which involves using a combination of analog and digital beamforming to reduce the complexity of the system. The analog beamforming is implemented using a set of phase shifters, which can be adjusted to steer the beam in different directions. The digital beamforming is implemented using the above-described linear combination of beam steering vectors.
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